Chapter 4 FLOWING FLUIDS AND PRESSURE VARIATION Fluid Mechanics Source:
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Transcript of Chapter 4 FLOWING FLUIDS AND PRESSURE VARIATION Fluid Mechanics Source:
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Chapter 4
FLOWING FLUIDS AND PRESSURE VARIATION
Fluid Mechanics
Source:
http://www.geofys.uu.se/files/teacher/
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Rotation and Vorticity
Rotation of a fluid element in a rotating tank of fluid(solid body rotation).
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Rotation of fluid element in flow between moving andstationary parallel plates
o You can think of the “plus signs” as small paddle wheels that are free to rotate about their center.
o If the paddle wheel rotates, the flow is rotational at that point.
Rotation: the average rotation of two initially mutually perpendicular faces of a fluid element.
The angle between the bisect line and the horizontal axis is the rotation, θ.
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As
And similarly
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The net rate of rotation of the bisector about z-axis is
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The rotation rate we just found was that about the z-axis; hence, we may call it
and similarly
The rate-of-rotation vector is
Irrotational flow requires (i.e., for all 3 components)
Applicable to ideal flow theory
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The property more frequently used is the vorticity, which is a vector equal to twice the rate of rotation vector
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Vortices
A vortex is the motion of many fluid particles around a common center. The streamlines are concentric circles.
Choose coordinates such that z is perpendicular to flow.In polar coordinates, the vorticity is (see p. 104 for details)
(V is function of r, only)
Solid body rotation (forced vortex):
or
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Vortex with irrotational flow (free vortex):
A paddle wheel does not rotate in a free vortex!
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In a cyclonic storm:Forced vortex (interior) andfree vortex (outside):
Good approximation to naturally occurring vortices such astornadoes.
Euler’s equation for any vortex:
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We can find the pressure variation in different vortices(let’s assume constant height z):
In general:
1) Solid body rotation:
2) Free vortex (irrotational):
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Application to forced vortex (solid body rotation):
with
Pressure as function ofz and r
p = 0 gives free surface